Preheat oven to 375F.
In a mixing bowl with a hand mixer or the bowl of a stand mixer fitted with the paddle attachment, mix the butter, shortening and confectioner's sugar on medium speed until light and fluffy, about 1 minute. Add the egg and extracts to the bowl, and mix to combine, scraping down the sides and bottom of the bowl.
Turn the mixer speed down to low and gradually add the flour and salt to the bowl, mixing until incorporated, about 30 seconds. Turn the mixer off, scrape down the bottom and sides and divide the dough in half. Remove one half of the dough from the bowl and add the red food color to the remaining half. Mix or knead it in until the color is blended into the dough evenly.
To shape the cookies, portion out about 1 teaspoon of each color of dough and roll them both into small logs. Pinch the logs together and give them a twist to create the spiraling effect of a candy cane. Place the shaped cookies onto an ungreased baking sheet, curving one end over to look like the hook of a candy cane. Once all the cookies are shaped, bake them until set and very light golden brown, about 9 minutes.
While the cookies are baking, mix together the crushed candy and granulated sugar. As soon as the cookies come out of the oven, sprinkle them with the candy sugar. Let the cookies cool on the tray for a few minutes so that the sugar can stick to them then transfer them to a rack to cool completely.